Download Modern Physics

Modern Physics
lecture X
Louis de Broglie
1892 - 1987
Wave Properties of Matter



In 1923 Louis de Broglie postulated that perhaps matter
exhibits the same “duality” that light exhibits
Perhaps all matter has both characteristics as well
Previously we saw that, for photons,
p
E hf h


c
c 

Which says that the wavelength of light is related to its
momentum

Making the same comparison for matter we find…

h
h

p mv
Quantum mechanics

Wave-particle duality



Waves and particles have interchangeable properties
This is an example of a system with complementary
properties
The mechanics for dealing with systems
when these properties become important is
called “Quantum Mechanics”
The Uncertainty Principle
Measurement disturbes the system
The Uncertainty Principle

Classical physics



Measurement uncertainty is due to limitations of the measurement
apparatus
There is no limit in principle to how accurate a measurement can
be made
Quantum Mechanics


There is a fundamental limit to the accuracy of a measurement
determined by the Heisenberg uncertainty principle
If a measurement of position is made with precision Dx and a
simultaneous measurement of linear momentum is made with
precision Dpx, then the product of the two uncertainties can never
be less than h/4p
DxDpx   / 2
Uncertainty principle
Energy and time
DE D   / 2
The Uncertainty Principle

In other words:


It is physically impossible to measure simultaneously the exact
position and linear momentum of a particle
These properties are called “complementary”


That is only the value of one property can be known at a time
Some examples of complementary properties are
 Which way / Interference in a double slit experiment
 Position / Momentum (DxDp > h/4p)
 Energy / Time (DEDt > h/4p)
 Amplitude / Phase
Erwin Schrödinger
1887 - 1961
Wave equations for probabilities

In 1926 Erwin Schroedinger proposed a wave
equation that describes how matter waves (or the
wave function) propagate in space and time
d
2m
  2 ( E  U )
2
dx

The wave function contains all of the information
that can be known about a particle
2

Wave Function


In quantum mechanics, matter waves are
described by a complex valued wave function, 
The absolute square gives the probability of
finding the particle at some point in space
   *
2

This leads to an interpretation of the double slit
experiment
Wave functions

The wave function of a free particle moving
along the x-axis is given by
 2px 
 x   A sin 
  A sin kx
  
This represents a snap-shot of the wave
function at a particular time
 We cannot, however, measure , we can
only measure ||2, the probability density

Max Born
Interpretation of the Wavefunction


Max Born suggested that  was the PROBABILITY
AMPLITUDE of finding the particle per unit volume
Thus
| |2 dV =   * dV
( * designates complex conjugate) is the probability of
finding the particle within the volume dV

The quantity | |2 is called the PROBABILITY
DENSITY

Since the chance of finding the particle somewhere in
space is unity we have
 ψ ψ* dV   ψ
2
dV  1
• When this condition is satisfied we say that the wavefunction
is NORMALISED
A particle or a wave?

( x )   A( ) sin
0
D small
and Dx big
Dpx small
2px

d
D big
Dx small
Dpx big and
Schrödinger Wave Equation




The Schrödinger wave equation is one of the most
powerful techniques for solving problems in
quantum physics
In general the equation is applied in three
dimensions of space as well as time
For simplicity we will consider only the one
dimensional, time independent case
The wave equation for a wave of displacement y
and velocity v is given by
 y 1  y
 2 2
2
x
v t
2
2
Solution to the Wave equation
We consider a trial solution by substituting
y (x, t ) =  (x ) sin(w t )
into the wave equation

2 y 1 2 y
 2 2
2
x
v t
• By making this substitution we find that
 2ψ
ω2
 2 ψ
2
x
v
• Where w /v = 2p/
and
• Thus
w 2/ v 2  (2p/)2
p = h/
Energy and the Schrödinger Equation
Consider the total energy
Total energy E = Kinetic energy + Potential Energy
E = m v 2/2 +U
E = p 2/(2m ) +U
 Reorganise equation to give
p 2 = 2 m (E - U )
ω 2 2m
 2 E  U 
 From equation on previous slide we get
2
v


• Going back to the wave equation we have
 2ψ 2m
 2  E  U ψ  0
2
x

• This is the time-independent Schrödinger wave
equation in one dimension
Solution to the SWE




The solutions (x) are called the STATIONARY
STATES of the system
The equation is solved by imposing BOUNDARY
CONDITIONS
The imposition of these conditions leads naturally
to energy levels
2


1
e
If we set U  

 4πε  r
0 

We get the same results as Bohr for the energy levels of the
one electron atom
The SWE gives a very general way of solving problems in
quantum physics
Probability and Quantum Physics






In quantum physics (or quantum mechanics) we
deal with probabilities of particles being at some
point in space at some time
We cannot specify the precise location of the
particle in space and time
We deal with averages of physical properties
Particles passing through a slit will form a
diffraction pattern
Any given particle can fall at any point on the
receiving screen
It is only by building up a picture based on many
observations that we can produce a clear
diffraction pattern
Wave Mechanics







We can solve very simple problems in quantum
physics using the SWE
This is sometimes called WAVE MECHANICS
There are very few problems that can be solved
exactly
Approximation methods have to be used
The simplest problem that we can solve is that of a
particle in a box
This is sometimes called a particle in an infinite
potential well
This problem has recently become significant as it
can be applied to laser diodes like the ones used in
CD players